Dottorato in SCUOLA DI SCIENZE STATISTICHE

Titolo della tesi: A multilinear approach to the theory of decision-making based on disaggregate and aggregate measures

This research work studies the criteria of rational choices being made by the decision-maker under conditions of certainty or uncertainty and riskiness. With regard to these choices, a same logical framework is shown. The criteria of rational choices being made by the decision-maker under conditions of certainty are based on non-negative and finitely additive masses, where each non-negative mass is associated with a possible alternative, and utility. The criteria of rational choices being made by the decision-maker under conditions of uncertainty and riskiness are based on probability and utility. Accordingly, this research work is connected with the international literature on the subject of probability viewed as a mass and preference. We study choices subjected to budget constraints being made by the decision-maker who is modeled as being a consumer. She chooses bundles of two goods, where each bundle operationally coincides with a bilinear measure of a metric nature. This measure is obtained from an estimated and summarized nonparametric distribution of mass. It is a joint distribution of mass. A bilinear measure is always decomposed into two linear measures obtained from two summarized nonparametric marginal distributions of mass. A joint distribution of mass together with its two marginal distributions of mass obey two different logical systems. Possible alternatives obey a logical system for which only two values are admissible. Non-negative masses associated with possible outcomes obey another logical system for which an infinite number of values is admissible. Such systems are strictly linked and superimposed. From observing the decision-maker’s choice behavior, it is possible to identify her subjective preferences as well as to estimate her subjective masses associated with possible alternatives. A nonparametric joint distribution of mass has always to reflect the knowledge hypothesis underlying each estimate concerning all conjoint masses characterizing it. This hypothesis is made by the decision-maker. Our goal is to extend rational choice behaviors. Our goal is to study multiple choices. They are associated with multiple goods. Each multiple choice is based on different estimated and summarized joint distributions of mass. Each multiple choice is rational if and only if all these summaries of joint distributions of mass are coherent. In Chapter 1, we define the notion of random good as well as the one of prevision bundle. We prove a theorem showing that there exists a full analogy between properties concerning average quantities of consumption of random goods and well-behaved preferences. We focus on axioms of revealed preference theory applied to average quantities of consumption of goods. Revealed preference theory gives empirical meaning to the neoclassical economic hypothesis according to which the best rational choice being made by the decision-maker inside of her budget set has to be the one maximizing her utility. We show that the best rational choice being made by the decision-maker inside of her budget set deals with average quantities of consumption of goods. The object of decision-maker choice under conditions of uncertainty and riskiness is a bundle of two random goods. After decomposing it inside of a two-dimensional Cartesian coordinate system, we define the decision-maker’s demand functions that give the average consumption amounts associated with each good under consideration. We show that it is possible to unify the empirical content of specific theories referred to coherent previsions of random goods in specific economic environments. In Chapter 2, we prove a theorem showing how to transfer all the n states of the world of a contingent consumption plan on a one-dimensional straight line on which an origin, a unit of length and an orientation are chosen. All the n states of the world of a contingent vi consumption plan are possible alternatives. They are not studied inside of E n only, where E n is an n-dimensional linear space over R having a Euclidean structure. This is because they are transferred on a one-dimensional straight line on which an origin, a unit of length and an orientation are established. Accordingly, we firstly observe a reduction of dimension because we pass from n to 1. Nevertheless, we continue to consider a finite number of points. Strictly speaking, we do not consider an n-dimensional point referred to a random good, where a random good identifies a contingent consumption plan, but we study a finite set of n one-dimensional points. We do not deal with n masses associated with n possible states of the world of a contingent consumption plan yet. If we focus on the two-good assumption then X1 and X2 are two random goods, where each of them has n possible values. Each of them has n possible alternatives. Each of them has n possible consumption levels. The n possible values for each good under consideration are transferred on two one-dimensional straight lines on which an origin, a same unit of length and an orientation are established. Such lines are the two axes of a two-dimensional Cartesian coordinate system. The space where the decision-maker chooses is her budget set. If we take her budget set into account then all masses associated with all possible consumption levels come into play. Her budget set is an uncountable subset of a two-dimensional linear space over R. Her budget set contains points whose number is infinite. It is a right triangle belonging to the first quadrant of a two-dimensional Cartesian coordinate system. The point given by (0,0) identifies its right angle, whereas the budget line whose slope is negative identifies its hypotenuse. Her budget set contains infinite coherent bilinear previsions associated with a joint random good denoted by X1 X2 and infinite coherent linear previsions associated with two marginal random goods denoted by X1 and X2. Two marginal random goods always identify a joint random good. Each bilinear prevision is denoted by P(X1 X2), where P(X1 X2) is always decomposed into two linear previsions denoted by P(X1) and P(X2) respectively. The decision-maker chooses one bilinear prevision denoted by P(X1 X2) among infinite coherent bilinear previsions. This is her rational choice. She chooses a bundle of two random goods operationally identified with P(X1 X2). Since P(X1 X2) belongs to a two-dimensional convex set, we express it in the form given by (P(X1), P(X2)). Accordingly, she also chooses P(X1) and P(X2) because P(X1 X2) is always decomposed into P(X1) and P(X2) respectively. It is clear that we secondly observe reductions of dimensions because we pass from 2 to 1. Indeed, we pass from P(X1 X2), where P(X1 X2) is found inside of a subset of a two-dimensional Cartesian coordinate system, to P(X1) and P(X2), where P(X1) and P(X2) are found on two different and mutually orthogonal one-dimensional straight lines. Two nonparametric marginal distributions of mass give rise to two continuous subsets of R, where each of them identifies a line segment belonging to one of the two axes of a two-dimensional Cartesian coordinate system. This is because all coherent previsions of marginal random goods are considered. All coherent previsions of two marginal random goods identify the two catheti of the right triangle under consideration. Such previsions are obtained by taking all values between 0 and 1, end points included, into account for each mass associated with a possible consumption level concerning a random good. The number of these values is infinite. A nonparametric conjoint distribution of mass gives rise to a continuous subset of R×R. This is because all coherent previsions of a joint random good are considered. They are obtained by taking all values between 0 and 1, end points included, into account for each mass associated with a possible value for two random goods which are jointly considered. The number of these values is infinite. We show that the continuous subset of R×R is a subset of the direct product of R and R, where the latter is a vii two-dimensional linear space over R. In Chapter 3, we define multiple goods of order 2 whose possible values are not necessarily of a monetary nature. We show a concrete example referred to a multiple physical good of order 2. Given the two-good assumption, the objects of decision-maker choice are studied by using bilinear measures of a metric nature. Such measures are firstly decomposed into two linear measures inside of the budget set of the decision-maker. We secondly establish aggregate measures which are strictly connected with multiple goods. Aggregate measures are based on what the decision-maker chooses inside of her budget set. They are studied outside of her budget set. The Cartesian product of two finite sets of possible quantities of consumption associated with two goods which are separately considered can be released from the notion of ordered pair of possible quantities of consumption connected with each good under consideration. This implies that an extension of the notion of bundle of goods is caught. Accordingly, we define the notion of consumption matrix. For the purpose, disaggregate and aggregate measures of a metric nature are considered. We calculate the average consumption as well as the variability of it associated with a multiple good of order 2. The variability of consumption is expressed by using the Bravais-Pearson correlation coefficient. We use the Bravais-Pearson correlation coefficient because the variability of a nonparametric joint distribution of mass is expressed by its numerator. This variability always depends on how the decision-maker estimates all the conjoint masses under consideration. She estimates them according to her variable state of information and knowledge. Accordingly, mean quadratic differences connected with multiple goods of order 2 can be shown. The Bravais-Pearson correlation coefficient associated with each bundle of two goods being chosen by the decision-maker inside of her budget set is used in order to check the weak axiom of revealed preference. We refer ourselves to this axiom because it is the basic axiom of the theory of decision-making whenever the decision-maker is modeled as being a consumer whose choices are subjected to budget constraints. We realize that a random good can always be studied by using a particular joint distribution of mass. Consumption data are dealt with by using metric measures. Disaggregate measures are obtained by using a linear and quadratic metric. Aggregate measures are obtained by using a multilinear and quadratic metric. In Chapter 4, we define a multiple random good of order 2 denoted by X12 whose possible values are of a monetary nature. A two-risky asset portfolio is a multiple random good of order 2. It is firstly possible to establish its expected return by using a linear metric. Given 1X and 2X, where 1X and 2X are the components of X12 = {1X, 2X}, whenever we use a linear metric in order to establish the expected return on a two-risky asset portfolio we focus on the components of X12 only. We secondly establish the expected return on X12 denoted by P(X12) by using a multilinear metric. Whenever we use a multilinear metric in order to establish the expected return on a two-risky asset portfolio we focus on X12. It is viewed as a stand-alone good. Whenever we use a multilinear metric we are not interested in studying separately the components of X12 denoted by 1X and 2X. If the decision-maker is risk neutral then P(X12) is a subjective price coinciding with the certainty which is judged to be equivalent to X12 by her. An extension of the notion of mathematical expectation of X12 denoted by P(X12) is carried out by using the notion of α-norm of an antisymmetric tensor of order 2. We prove a theorem about this. An extension of the notion of variance of X12 denoted by Var(X12) is shown by using the notion of α-norm of an antisymmetric tensor of order 2 based on changes of origin. We prove a theorem about this. An extension of the notion of expected utility connected with X12 is considered. An extension of Jensen’s viii inequality is shown as well. Whenever the decision-maker maximizes the expected utility of X12 she maximizes the utility of average quantities of consumption. We focus on how the decision-maker maximizes the expected utility connected with multiple random goods of order 2 being chosen by her under conditions of uncertainty and riskiness. What she actually chooses inside of her budget set underlies all of this. In Chapter 5, we study m risky assets identifying a multiple random good of order m whose possible values are of a monetary nature. Any two risky assets of m risky assets are always studied inside of the budget set of the decision-maker. Two or more than two risky assets are also studied outside of her budget set. Whenever changes of origin are considered we go away from her budget set. Given m risky assets subjected to m changes of origin, we study an m-dimensional linear manifold embedded in E n . It is spanned by m basic risky assets, where each of them is subjected to a change of origin. Each of them has n possible values. Each of them has n possible alternatives. Each linear combination of m basic risky assets identifies an n-dimensional vector belonging to an m-dimensional linear manifold embedded in E n , where this n-dimensional vector is a risky asset. This n-dimensional vector identifies a nonparametric marginal distribution of mass. The number of all linear combinations of m basic risky assets is infinite. All risky assets belonging to an m-dimensional linear manifold embedded in E n are dealt with. We are also interested in knowing the starting possible values for each risky asset under consideration as well as all marginal masses associated with them. We show that all risky assets contained in an m-dimensional linear manifold embedded in E n are intrinsically related. In particular, we realize that any two risky assets of them are α-orthogonal, so their covariance is equal to 0. We define the notion of α-metric tensor. It is used to study how all risky assets contained in an m-dimensional linear manifold embedded in E n are intrinsically related. On the other hand, eigenvalues, eigenvectors, eigenequation and eigenspaces derive from the notion of α-metric tensor. We show that all principal components coincide with basic risky assets. Constants of riskiness explain the variance of all risky assets belonging to an m-dimensional linear manifold embedded in E n . We show that all risky assets belonging to a specific m-dimensional linear manifold embedded in E n are proportional. Non-classical inferential results are obtained. We realize that the price of risk is based on multilinear indices. This price measures how risk and return can be traded off in making portfolio choices.